#### by Yves Meyer

The letters which are displayed here provide a breathtaking diary of
Ingrid Daubechies’ scientific achievements on wavelets.

The wavelet revolution blew up in the early eighties. It created an
immense enthusiasm and an unusual understanding between scientists
working in completely distinct fields. We believed that barriers in
communication would disappear for ever. We thought we were building a
united science. How did this start? It began in the sixties. David
Hubel and Torsten Wiesel had discovered that some proto-wavelets were
seminal in the functioning of the primary visual cortex of mammals. In
his famous book (*Vision*, MIT Press, 1982) David Marr proposed a fascinating
algorithm which could emulate this processing.
Jacques Magnen,
Roland Sénéor,
and
Kenneth Wilson,
were using some other proto-wavelets
in quantum field theory. Phase space localization was the favorite
tool of
James Glimm
and
Arthur Jaffe.
Unfolding a signal in the time
frequency plane was proposed by
Eugene Wigner
in the thirties.
Wigner’s motivation was quantum mechanics. Wigner was followed by
Dennis Gabor
and
Claude Shannon
in the forties. In mathematics
Alberto Calderón
had already introduced a variant of a wavelet analysis as
an alternative to Fourier analysis. In the late seventies wavelets
were popping up everywhere. But the needed unification was still
missing until the French geophysicist Jean Morlet (1931–2007) made the
fundamental discovery which is described now. Morlet was working for
the Oil company Elf Aquitaine (now Total) and his research was
motivated by problems encountered in analyzing some reflected seismic
waves. These seismic waves were a key tool in oil prospecting. Working
on these specific signals, Morlet
experimented with the flaws of windowed
Fourier analysis. Then
he introduced a revolutionary method which
would later be named “wavelet analysis” and observed that this new
tool was more efficient than what was used before. Twenty years later
he was awarded the Reginald Fessenden Prize. In 1997 Pierre
Goupillaud introduced Jean Morlet with these words:

Morlet performed the exceptional feat of discovering a novel mathematical tool which has made the Fourier transform obsolete after 200 years of uses and abuses, particularly in its fast version…Until now, his only reward for years of perseverance and creativity in producing this extraordinary tool was an early retirement from ELF.

Goupillaud was right. Soon after his discovery Morlet was fired by ELF and suffered a breakdown. At that time Morlet had a vision without a proof and needed some help. Alex Grossmann (1930–2019), a physicist at the Centre de Physique Théorique, Marseilles-Luminy, listened to Morlet with much patience and sympathy. Finally Alex understood Morlet’s claim and gave it the sound mathematical formulation which is described now.

Let __\( s(t) \)__ be a signal with a finite energy. Its wavelet transform
__\( W(a, x) \)__ is a continuous function of two variables __\( a > 0 \)__ and __\( x\in
\mathbb{R} \)__. This wavelet transform depends on the signal __\( s(t) \)__ but
also on the wavelet __\( \psi \)__ which is used in the analysis. How much the
choice of this wavelet will affect the results is an important
problem. Before discussing it let us return to the definition of the
wavelet transform. In this definition __\( 1/a \)__ is the magnification of
the “mathematical microscope” __\( \psi \)__, and __\( x \)__ is a real number
indicating the place where you want to zoom into the signal __\( s(t). \)__
Finally __\( W(a, x) \)__ is the cross-correlation __\( \langle s, \psi_{a, x}\rangle \)__ between
the signal __\( s \)__ and the shrunk or dilated wavelet
__\[
\psi_{a, x}=\frac{1}{a}\psi\biggl(\frac{t-x}{a}\biggr).
\]__
We
have
__\[
W(a,x)=\int_{-\infty}^{+\infty} \overline{\psi_{a, x}}(t)s(t)\,dt.
\]__
The function __\( \psi \)__ is named the
*analyzing wavelet* (the microscope)
and the overline means complex conjugate. This analyzing wavelet
shall be a smooth function, localized around 0, and oscillating.
This last requirement means __\( \int_{-\infty}^{+\infty} \psi(t)\,dt=0. \)__
Finally __\( \psi \)__ shall be an admissible wavelet: both
__\[
\int_0^{\infty}|\widehat \psi(u)|^2\,\frac{du}{u}=1 \quad
\text{ and }
\quad
\int_0^{\infty}|\widehat \psi(-u)|^2\,\frac{du}{u}=1
\]__
are needed.
Here __\( \widehat \psi \)__ is the Fourier transform of __\( \psi \)__. Any function
__\( \psi \)__ which is real valued, smooth, localized, and oscillating
becomes an admissible wavelet after multiplication by a suitable
constant.

Then Grossmann and Morlet proved that every signal __\( s(t) \)__ can be
exactly reconstructed by a simple inversion formula.
(See A. Grossmann and J. Morlet, “Decomposition of Hardy functions into square integrable wavelets of constant shape,”
*SIAM J. Math. Anal.* **15** (1984) 723–736.) Everything
works as if the analyzing wavelets __\( \psi_{a, x} \)__ were an orthonormal
basis. Indeed under the assumption that __\( \psi \)__ is an admissible
wavelet we have
__\[ s(t)=\int_0^{\infty}\!\!\!\int_{-\infty}^{+\infty}W(a, x)
{\psi_{a, x}}(t)\,dx\,\frac{da}{a}.
\]__

As it was announced by Morlet, wavelet analysis provides us with a
better understanding of signals which cannot be analyzed correctly by
a standard windowed Fourier analysis (Gabor wavelets, 1946).
This is the
case when strong transients occur in the signal. It is also the case
for fractal or multifractal signals, as
was proved by
Alain Arneodo,
Uriel Frisch,
Giorgio Parisi
and their collaborators.
Stéphane Jaffard
proved that the choice of the analyzing wavelet was
not very important as long as __\( \psi \)__ is sufficiently smooth, well
localized and has enough vanishing moments. The wavelet
__\( \psi(t)=\cos(5t)\exp(-t^2/2) \)__ (Morlet’s favorite) does not meet this third
condition. The integral of __\( \psi \)__ is extremely small but does not
vanish.

The continuous wavelet analysis of a signal __\( s(t) \)__ is highly
redundant. For some analyzing wavelets __\( \psi \)__ the wavelet transform
__\( W(x, a) \)__ of a signal __\( s(t) \)__ is the solution to a partial differential
equation. The analysis of a signal of length __\( N \)__ produces __\( N^2 \)__
coefficients. In some cases this redundancy can be useful and in other
cases the computational load is prohibitive. At the other extreme of
the picture we find the Fast Wavelet Transform which is described
below.

The story I am telling now began on January 15, 1985, when I met Alex Grossmann in Marseilles for the first time. A month earlier Jean Lascoux, a physicist and a colleague at Ecole Polytechnique, had given me a fascinating preprint by Grossmann and Morlet. This preprint was so attractive that I could not resist traveling to Marseilles. I spent three days talking with Alex and I soon became his disciple. I shared his values and ethics. Among these values I would single out an intense curiosity, a great humility, a profound confidence in others and an outstanding capacity for friendship. Alex became a spiritual father and a scientific guide.

In a joint work with Alex and Ingrid we constructed a frame of
__\( L^2({\mathbb R}^n) \)__ of the form
__\[
\psi_{j, k}(x)=2^{jn/2}\psi(2^jx-k),\quad j\in {\mathbb Z},\,k\in {\mathbb Z}^n,
\]__
where the analyzing wavelet __\( \psi \)__ belongs to the Schwartz class. The
wavelet coefficients of a signal __\( s(t) \)__ are __\( c(j,k)=\langle s, \psi_{j,k}\rangle. \)__
These coefficients are still redundant but the reconstruction of the
signal __\( s(t) \)__ is exact. This __\( \psi \)__ is named the mother wavelet since
it generates the other wavelets in the frame. Our goal was to build a
digital version of wavelet analysis which would be exact. Our work was
entitled “Painless nonorthogonal expansions” and was published in *J. Math. Phys.* **27** (1986), 1271–1283. I had not yet met Ingrid at that
time. This explains why in
[1]
Ingrid is eager to meet.

However I was dissatisfied with this result and I wondered if, instead
of a frame, I could construct a true orthonormal basis with the same
structure. During the Summer of 1985 I did it and I mailed the
manuscript to Ingrid. In the Fall of 1985 I gave a lecture at the
Courant Institute on this construction. On December 3, 1985, Ingrid
acknowledged receipt of my draft
(letter
[1]
and apologized
for missing my talk. My construction was published six months later as
“Principe d’incertitude, bases hilbertiennes et algèbres
d’opérateurs” (The uncertainty principle, Hilbert base and operator
algebras) in *Séminaire Bourbaki*, 38-ème année, February 1986,
vol. 1985/86). Using this basis every Schwartz distribution __\( S \)__ is
encoded as a simple sequence __\( c_k,\,k=0, 1, \dots \)__ of numbers and the
intricate properties of __\( S \)__ become simple growth conditions on this
sequence __\( c_k. \)__ For a functional analyst this was paradise. However
such wavelets had infinite supports and were useless in real life
problems. One year later Ingrid overcame that obstacle and constructed
orthonormal wavelet bases which are smooth (__\( m \)__-continuous
derivatives) and compactly supported. The mother wavelet depends on
the required regularity __\( m \)__
[6].
When compared with the
deepness of Ingrid’s achievement my construction looks trivial. In her
letter of July, 1986,
[2]
Ingrid wanted to know whether my
construction of an orthogonal wavelet basis could be modified in such
a way that only fine scales would play a role. She proposed a way for
doing it. In a joint work with
P-G. Lemarié
I had already met this
goal. This was published as “Ondelettes et bases hilbertiennes,” *Rev. Mat. Iberoam.* **2**:1-2, (1987), 1–18.

In November 1986 I was invited to give a talk at the University of
Chicago. It was the time when
Stéphane Mallat
was writing his Ph.D.
at the Department of Computer and Information Science of the
University of Pennsylvania. Stéphane was eager to
talk with me
and we met at Eckhart Hall. Stéphane went there with some breaking
news. He had discovered that the construction of orthonormal wavelet
bases was obeying the same algebraic rules as the design of quadrature
mirror filters. These filters were already a subject of intense
investigation in electrical engineering. Stéphane was bridging the
gap between mathematics and electrical engineering. We had three days
of intense work in
Professor Zygmund’s office.
Antoni Zygmund
was still alive but let us freely use his office. My main contribution
to this discussion was to warn Stéphane that the iterative procedure
yielding wavelets could diverge when applied to some “bad” quadrature
mirror filters (see D. Esteban and C. Galand, “Application of
quadrature mirror filters to split band voice coding schemes,” *Proc.
IEEE ICASSP* (1977), 191–195). This point was going to be fully
understood a few years later by
Albert Cohen
and Ingrid. Moreover we
were unable to characterize the “good” quadrature mirror filters that
yield smooth wavelets after iteration. A few months later Ingrid
solved these two problems and achieved her splendid construction of
orthonormal bases of compactly supported smooth wavelets
([4]
and
[5]
written two days later). In
[4]
and
[5]
Ingrid explains her constructive vision of compactly
supported scaling functions and wavelets. This vision was consistent
with the framework introduced by Mallat and
me, while bringing
completely novel insight to the needed algebraic manipulations to
obtain the filter banks that ensure both compact support and
smoothness on the resulting wavelets. A month later Ingrid sent me the
magnificent letter
[6]
where she describes her construction
of orthonormal wavelet bases with compact support.

As indicated in her letter
[4]
Ingrid was concerned with a
problem raised by scientists working in computer vision. They wanted
some even (or odd) wavelets while Ingrid had proved that the Haar
system is the only compactly supported orthonormal wavelet basis with
this property. That is why Ingrid and Albert Cohen constructed smooth
and symmetrical biorthogonal wavelets (the ones which are used in the
compression standard JPEG2000). I was eager to publish their beautiful
paper in *La Revista Matemática Iberoamericana* and Ingrid sent
me three copies for submission to this journal. It was a time when
e-mail did not exist and a paper version of an article was still
needed to submit it.
In
[8]
(dated December 1, 1987) Ingrid says how much she is sorry for the death of my mother.
Let me say today how much I was so moved by your kind letter, dear
Ingrid. In
[9]
(January 15, 1990) Ingrid acknowledges receipt
of my book and begins discussing time-frequency wavelets.

The panorama dramatically changed around 1990. Motivated by a problem
raised by Kenneth Wilson, Ingrid, in a joint work with Stéphane
Jaffard and
Jean-Lin Journé,
constructed an orthonormal basis of
time-frequency wavelets. The problem solved by Ingrid et al. has a
long history. It was raised by Dennis Gabor in 1946. Gabor thought
that the functions
__\[
\exp(2\pi i k x)\exp(-(x-l)^2/2),\quad k,l\in{\mathbb{Z}},
\]__
could be a basis of __\( L^2({\mathbb{R}}) \)__. When he
raised this issue
Gabor anticipated the digital revolution. He tried to find an
efficient way to encode a speech signal. He believed that a speech
signal __\( s(x) \)__ could be encoded as a short sequence of numbers __\( c(k,
l),\,k,l\in{\mathbb Z} \)__. These numbers would have been the
coefficients of __\( s(x) \)__ in the Gabor basis. Unfortunately Gabor was
wrong and the Gabor basis is not a basis. Something was missing. It
is like omitting the number 7 in
arithmetic. Ingrid could fix the
problem and her discovery (a joint work with Jaffard and Journé)
happened to be seminal in the detection of gravitational waves
(see Sergey Klimenko et al., “Observing gravitational-wave
transient GW150914 with minimal assumptions” at https://dcc.ligo.org/LIGO-P1500229/public/main).

In
[11]
Ingrid is interested in the application of wavelets
to scientific computing. She discusses a new algorithm discovered by
Greg Beylkin,
Ronald Coifman
and
Vladimir Rokhlin.
The goal is to
accelerate the computation of the product of two large __\( N\times N \)__
matrices under the hypothesis that these matrices are almost
diagonal in a wavelet basis. The charming letters
[12]
and
[13]
offer a moving picture of a pregnant Ingrid.

The letter
[16]
is not dated but I would say that it was
written circa 1992 since Ingrid’s daughter Caroline begins to walk.
Soon after the construction by Ingrid et al. of orthonormal bases of
time-frequency wavelets Coifman and I found another solution in which
the time segmentation could be arbitrarily imposed. In
[16]
Ingrid with her usual fairness stresses that we should mention the
contributions of H. Malvar, J. P. Princen, and A. B. Bradley who
anticipated our construction under the name of * lapped
transforms*. And Martin Vetterli should not be forgotten!
Orthonormal bases of time-scale wavelets previously existed under the
name of quadrature mirror filters, as Mallat discovered. Similarly
orthonormal bases of time-frequency wavelets already existed as
lapped transforms. Analysts should be modest. But in
[16]
Ingrid is mostly discussing the construction of an orthonormal
wavelet basis on a given interval. Everything needs to be localized
on this specific interval, which was not the case for the preceding
constructions. This paves the way to the problems discussed in the
letter
[17].

The letter [17] is extremely interesting and exemplifies the differences between Ingrid and me. Ingrid is at the same time a mathematician, a physicist and a scientist. She immediately perceived the talent of Wim Sweldens. Ingrid wished that Wim would be hired by Princeton. Princeton asked for my advice. Before giving my opinion I wrote several emails to Ingrid. Ingrid finally answered but her mail broke down while she was writing. This was fortunate and we have this long and beautiful letter. Ingrid discusses the relevance of mathematics in programming and states that efficient algorithms shall be based on good and deep mathematics. This is a marvelous statement. Finally Wim moved to high tech and is famous for several spectacular innovations.

Ingrid’s letters offer a genuine insight into this exceptional decade in which wavelet analysis was elaborated. In these letters Ingrid is actively present with her scientific vision, her enthusiasm, her doubts, her fairness, and her extraordinary ability to communicate. Ingrid trusted me and never hesitated to unveil her deepest thoughts and feelings. Thanks, Ingrid